Manufacturing processes for producing products usually rely on quantitative measurements to provide information required for process control. Such measurements can be made on the final product, and/or on intermediate stages of the product within the manufacturing process, and/or on tools/fixtures used in the manufacturing process. For example, in semiconductor chip fabrication, measurements can be performed on finished chips (i.e., final product), on a wafer patterned with a photoresist (i.e., intermediate stage), or on a mask (i.e., a tool or fixture). Frequently, as in the case of semiconductor chip fabrication, these measurements are performed on structures having small dimensions. Furthermore, it is highly desirable to perform process control measurements quickly and non-destructively, in order to ensure a minimal impact on the process being controlled. Since optical measurements can be performed quickly, tend to be non-destructive, and can be sensitive to small features, various optical process control measurements have been developed.
Optical process control measurements can often be regarded as methods for measuring parameters of a pattern. For example, a pattern can be a periodic one-dimensional grating of lines on the surface of a wafer, and the parameters to measure can be line width, line spacing and depth of the grating. To measure these parameters, an optical response of the pattern is measured. For example, reflectance as a function of wavelength can be measured. Typically, the optical response will depend on the parameter (or parameters) of interest in a complicated way such that direct parameter extraction from measured data is impractical. Instead, a mathematical model is typically constructed for the pattern, having the parameters of interest as variables. Within the model, a modeled optical response is calculated corresponding to the measured optical response. The parameters of interest are then determined by adjusting the variables to fit the modeled response to the measured response. Various optical process control measurements differ depending on the measured response(s), and on the kind of mathematical model employed.
In particular, the mathematical model employed for such measurements can be more or less rigorous. Generally, more rigorous models provide improved fidelity between measured and modeled results, but require greater calculation time and/or processing resources. In addition, rigorous models can require more detailed measurements, which tends to increase measurement time and/or cost. Less rigorous models reduce calculation time and/or required processing resources, but tend to provide reduced fidelity between measured and modeled results.
A rigorous modeling approach described by Moharam et al. in Journal of the Optical Society of America (JOSA), A12, n5, p 1068–1076, 1995 is known as the rigorous coupled wave analysis (RCWA). The RCWA is limited to periodic structures such as a grating, and the required computation time is generally large, especially for a grating having a period substantially larger than a wavelength. Rigorous modeling approaches other than the RCWA have also been developed, but such approaches have similar advantages and drawbacks as the RCWA. The RCWA was first introduced by K. Knop in JOSA, v68, p 1206, 1978, and was later greatly improved by Moharam et al. in the above-referenced article. Some implementations of the RCWA are described in U.S. Pat. Nos. 6,590,656 and 6,483,580 assigned to KLA-Tencor, U.S. Pat. No. 5,963,329 assigned to IBM, and U.S. Pat. No. 5,867,276 assigned to Bio-Rad.
An example of a less-rigorous modeling approach is referred to herein as the scalar model. The basic idea of the scalar model is to divide the pattern of interest into several features, calculate an optical response (e.g., complex amplitude reflection or transmission coefficient) of each feature in a plane-wave approximation (i.e., as if each feature had infinite lateral extent), and then combine the calculated plane-wave responses of each feature to obtain an approximate modeled response for the pattern. For example, if features 1 and 2 have plane-wave complex amplitude reflection coefficients r1 and r2 respectively, where r1 and r2 are referred to the same reference plane, then the combined reflectance R in the scalar model is given by R=|a1r1+a2r2|2, assuming lateral coherence. Here a1 and a2 are the areal fractions of features 1 and 2 respectively (i.e., the fraction of the total pattern area in features 1 and 2 respectively). Thus diffraction is ignored in the scalar model, but interference between features can be accounted for.
The scalar model was originally proposed by Heimann et al. (Journal of the Electrochemical Society, v131, p 881, 1984; ibid, v132, p 2003, 1985). Applications of the scalar model are considered by Cho et al. (Journal of Vacuum Science and Technology (JVST), B20, p 197, 2002), Maynard et al. (JVST, B13, p 848, 1995; JVST, B15, p 109, 1997), and Lee et al. (International Conference on Characterization and Metrology for ULSI Technology, Gaithersburg, Md., Mar. 23–27 1998, AIP conf. proc., v449, p 331, 1998). Aspects of the scalar model are considered in U.S. Pat. Nos. 6,281,974 and 6,100,985 by Scheiner et al., assigned to Nova Measuring Instruments Ltd.; U.S. Pat. Nos. 6,623,991 and 6,340,602 by Johnson et al. assigned to Therma-Wave Inc. and Sensys Instruments respectively; and U.S. patent application Ser. No. 10/607,410 by Li et al. entitled “Method and Apparatus for Examining Features on Semi-Transparent Substrates” and assigned to n&k Technology Inc.
This conventional scalar model works well for large features (i.e., features substantially larger than an optical wavelength), but its accuracy decreases with feature size, and it typically does not provide sufficiently accurate results for features having a size on the order of a wavelength or smaller. However, the scalar model is much simpler than rigorous modeling approaches, such as the RCWA, and thus requires far less computation time and/or processor resources. Accordingly, it would be an advance in the art to improve the accuracy of the scalar model for features having small size.
Thus an object of the present invention is to provide a model having improved accuracy compared to the scalar model for patterns having small feature sizes. A further object of the invention is to provide a model having reduced computation time compared to rigorous modeling approaches, such as the RCWA. Yet another object of the invention is to achieve the preceding two objects simultaneously.